Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem

in a viscosity sense, and in a pointwise sense at regular free boundary points. Here ν is the inward unit spacial normal to the free boundary D∩ ∂{u > 0}, u = max(u,0) and u− = max(−u,0). This work continues the study in [C1], [C2] and [CLW], where uniform estimates for uniformly bounded solutions to Pε were obtained; these estimates allow the passage to the limit, as ε→ 0, uniformly. As in the present paper, [C1], [C2] and [CLW] are concerned with the two phase case, that is, the functions u are allowed to change sign and become negative. Also the approach is local, since the solutions u are not forced to be globally defined nor to take on prescribed initial or boundary values. The problem of studying uniform properties and the limit as ε → 0 of solutions to Pε had been previously considered—in the one phase case—in [BCN] and in [CV]. This problem appears in combustion in the description of laminar flames as an asymptotic limit for high activation energy (see [BL]). We refer to [V] for a survey of related results.